The circulation in the Black Sea characterized by a strong basin-wide
current along the shore is subject to mesoscale variability in the
form of meanders, intense jets, eddies and filaments. We present the
results of a new series of laboratory experiments where the typical
features of the circulation were modeled. The dynamical similarity of
the important dimensionless parameters governing the dynamics of the
system was satisfied in our experiments. The results demonstrate the
development of the baroclinic instability due to the fresh water
discharge imitating the river inflow in the Black Sea. Persistent
transient features of the circulation, such as the so-called Batumi
Eddy and the Sevastopol Eddy as well as other features are reproduced
in our experiments when the background rotation rate of the system is
varied. Quantitative data were obtained for the observed velocity and
vorticity fields. The results of the experiments are in agreement with
observational data as well as numerical modeling.

Thrombin is best known for its role in the chemical reactions leading
to the formation of fibrin, one of the structural components of a
blood clot. The production of thrombin involves two main pathways,
one of which requires positive feedback by thrombin. A mathematical
model of thrombin generation in the absence of this feedback pathway
was developed to investigate the production of thrombin in human
ovarian follicular fluid, an avascular system where the presence of
most of the coagulation proteins has recently been discovered.
This model consists of 54 nonlinear ordinary differential equations
describing the concentrations of the various proteins and lipid
involved. Simulations of the model equations help to determine the
controlling factors in restricting thrombin levels in follicular
fluid, as well as suggest various laboratory experiments that should
be conducted to elucidate the role of thrombin in this system.
Applications of this work include determining a probable role for
thrombin in folliculogenesis and the implications of novel
anticoagulant therapies on fertility.

The dynamic nature of nuclear architecture within eukaryotic cells is
an interesting problem in cellular biology. Specifically, the origin,
maintenance and disappearance of speckles, which are heterogeneously
distributed nuclear compartments enriched with pre-mRNA splicing
factors, is unknown. It has been hypothesized that a process of
self-aggregation among dephosphorylated splicing factors, modulated by
a phosphorylation-dephosphorylation cycle, is responsible for the
formation and disappearance of speckles. Also, it is thought that the
existence of an underlying nuclear structure plays a major role in the
organization of splicing factors.

We will explain how these hypotheses and a diffusion-approximation
approach allow for the derivation of a fourth order
aggregation-diffusion model that describes a possible mechanism
underlying the organization of splicing factors in speckles. A linear
stability analysis, supplemented with numerical simulations, will show
how the self-interaction among dephosphorylated splicing factors can
result in spatial patterns that are caused by instabilities about
homogeneous steady states.

Directed cell movement in response to an external chemical gradient is
involved in many diverse physiological processes such as wound healing
and embryogenesis. The small G proteins, a class of proteins known to
regulate actin polymerization, establish intracellular concentration
gradients in response to external stimuli. Enhanced polymerization of
the protein actin at the leading edge of the cell generates sufficient
force for membrane protrusion and movement. Using mathematical
modelling, we simulate the orientation and density of actin filaments
and polymerizing filament tips against a rigid obstacle. We compare
our results to experimental data and discuss various hypotheses
regarding the formation of the actin network and regulation of actin
polymerization at the leading edge of a motile cell.

We developed a new reconstruction method called the inverse polynomial
reconstruction method for the resolution of the Gibbs phenomena. It is
well known that the Fourier approximation of nonperiodic or
discontinuous function or the polynomial approximation of
discontinuous function in a given interval suffers from the so-called
Gibbs phenomena. The convergence of the Fourier approximation in the
smooth region is O(1/N) and O(1) in the neighborhood of
discontinuity in its maximum norm due to the Gibbs oscillations.

The proposed inverse method seeks a polynomial reconstruction such
that the residue between the Fourier representations of the
reconstructed function and the original function is orthogonal to the
Fourier space. Consequently the reconstrucion is uniquely determined
and the method yields spectral accuracy removing the Gibbs
oscillations completely. Furthermore the reconstruction is exact if
the original function is a (piecewise) polynomial. We provide some
numerical examples for both one and two dimensional problems.

This is joint work with Bernie D. Shizgal at the University of British
Columbia.

We present stability of switching systems with time delay by using
Lyapunov Functional and Lyapunov Function. We extend results of ODE
switching systems to stabilize a class of DDE switching systems given
a proper condition on the time delay.

Far field wake behind a towed self-propelled body is simulated by
using a point force/force doublet idealization. These wakes become
unstable and form vortex streets. A new series of high-resolution 2D
numerical simulations is performed to study the characteristics of the
wakes including the shedding frequency for a wide range of control
parameters such as translational velocity, magnitude and spatial
extent of a localized force.

The results of numerical experiments of unsteady wake flow show the
existence of a great variety of flow regimes and are in good
qualitative agreement with preliminary laboratory experiments.

The particle boundary layer (PBL), a new terminology, is defined as
the region in which the concentration profile of particles has a
noticeable variation in depth. Although particle-driven gravity
currents are commonly modelled by assuming a vertically well-mixed
particle distribution, such PBL exists theoretically, especially for
larger size of particles. Similar observations are also obtained from
experiments; a highly stratified density profile of a
low-concentration flow has been observed, particularly near the
surface of solid bodies such as a river bed. To fully understand the
dynamics of the flow within the PBL, of interest will be constructing
a mathematical model to investigate how the particle distribution and
scalings in the governing equations are affected within such a layer.
In addition, the typical thickness of the PBL can be approximated
algebraically from the transport equation, which is then compared with
that of the viscous boundary layer. The ratio of the thickness of
the two layers eliminates the viscous terms from the Navier-Stokes
equations resulting in a system of first-order partial differential
equations. Theoretical solutions of the system will be shown and
discussed in several cases using perturbation theory and method of
characteristics. These results are then compared with numerical
solutions obtained using finite-difference scheme.

We use a simple compartmental model to show a possible mechanism for
multiple outbreaks or even sustained periodic oscillations of emerging
infectious diseases due to the psychological impact of reported
numbers of infective and hos
pitalized individuals. This impact leads
to the change of avoidance and contact patterns at both individual and
community levels, and incorporating this impact using a simple
nonlinear incidence function into the model shows qualitative
differences of the transmission dynamics.

A probability density function (PDF) fumigation model is presented
here to study the dispersion of air pollutants emitted from a tall
stack on the shoreline. This work considers dispersion of the
pollutants in the stable layer and within thermal internal boundary
layer (TIBL) proceeds independently. The growth of TIBL is considered
parabolic with distance inland and turbulence is taken as homogeneous
and stationary within the TIBL. Dispersion of particles (contaminant)
in lateral and vertical directions is assumed independent of each
other. This assumption allows us to consider the position of particles
in both directions as independent random variables. The lateral
dispersion distribution within the TIBL is considered as Gaussian and
independent of height. A skewed bi-Gaussian vertical velocity (w) PDF
is used to account for the physics of dispersion due to different
characteristics of updrafts and downdrafts within TIBL. Incorporating
finite Lagrangian time scale for the vertical velocity component, it
is observed that it reduces the vertical dispersion in the beginning
and moves the point of maximum concentration further downwind. Due to
little dispersion in the beginning, there is more plume to be
dispersed causing higher concentrations at large distances. The model
has considered Weil and Brower's (1984) convective limit to analyze
dispersion characteristics within TIBL. The revised model discussed
here is evaluated with the data available from the Nanticoke field
experiment on fumigation conducted in the summer of 1978 in Ontario,
Canada. The results of the revised model are in better agreement with
the observed data. The paper suggests the use of mean absolute error
and mean relative error as quantitative measures of model performance
along with the residual analysis.

We probe the micron-scale rheology of Carbopol ETD2050, a polymer with
a wide range of industrial applications, by studying the motion of
small suspended tracer particles. With the aid of video fluorescence
microscopy and particle tracking software, we record the positions of
several tracer particles simultaneously and extract the local
viscoelastic properties of the fluid from their dynamics. We study
these properties as a function of polymer concentration and tracer
particle diameter.

Accurate assessment of the volume of cerebral ventricles on computed
tomographic images of the brain is an important and as yet unsolved
problem in neuroradiology. Current subjective assessment of
ventricles by neuroradiologists and neurosurgeons has limited
accuracy, because of the complex shape of the ventricular system. We
are developing an automated system that can segment the cerebral
ventricles on axial computed tomographic images of the brain. Two
automated segmentation techniques have been developed and tested. One
is based on thresholding and the other on region growing. The results
have been compared to a manual segmentation by calculating the
similarity index (S). A good result (S > 0.7) was obtained.

This poster will present new results regarding symmetric binary
fractal trees, obtained using methods of computational topology.
Fractal trees are interesting mathematical objects and they have many
biological applications. A symmetric binary fractal tree, as first
introduced by Mandelbrot, has two parameters that define the
branching: the scaling ratio r (with 0 < r < 1) and a branching
angle q (with 0 < q < 180). We study the structure of
the self-avoiding and self-contacting trees with an analysis of the
corresponding e-neighbourhoods (the tree along with points
that are with e). The homology of the
e-neighbourhoods changes as e® 0, and depends on
the values of r and q. This provides new classifications of
the trees, and new topological critical points. We also compare
growth rates of holes with the scaling dimension.

In a variety of physical phenomena, we often want to track the motion
of an interface. Such phenomena can occur in fluid mechanics, material
science, medical science, control theory, and image processing. In
this poster, we introduce, analyze, and utilize level set methods for
the study of such problems. Level set methods are powerful numerical
techniques for tracking the motion of interfaces moving in complex
ways. They are based on computing solutions to approximate the
equations of motion of the fluid. They use techniques borrowed from
hyperbolic conservation laws.

One third of the power transmission line outages are caused by extreme
weather in Alberta. The lightning is one of the major adverse weather
conditions that can cause frequent power system outages. However, the
characteristics of lightning and the relationship between lighting and
power system outages are still not fully understood by power system
planning engineers. This research is to reveal the spatial and
temporal patterns of lightning in Alberta. It is found that both the
spatial and temporal distributions of the lightning are not
uniform. The area with the highest lightning frequency and the highest
lightning occurrence days are located at central Alberta along Rocky
Mountains. And the lightning frequencies and the lightning occurrence
days have decreasing trends from this center to the other parts of
Alberta. Meanwhile, the geographical and temporal characteristics of
lightning-caused transmission line outages on several voltage levels
are being studied. The underlying distribution of outage duration
hours on each voltage level are significantly different so that any
statistical analysis of the outages has to be carried out separately
for each of the voltage levels. Furthermore, linear models between the
outage duration hours and the lightning weather elements, by using the
best subset selection procedure and Jackknife cross validation method,
were established.

Computer modeling of complex phenomenon now plays an important role in
all areas of science and engineering. It is often the case that such
models are based upon complicated systems of differential
equations. MIRKDC [Enright, Muir] is a software package for the
numerical solution of systems of first order, nonlinear, boundary
value ordinary differential equations, with separated boundary
conditions. My research involves modification of the MIRKDC software
package in order to incorporate a number of significant performance
enhancements including analytic derivative assessment, computational
derivative approximation, problem sensitivity (conditioning)
assessment, defect control improvement and an auxiliary global error
indicator.

Contact rate and quarantine rate in the modeling of infectious
diseases are very important. Yet it is always not easy to identify the
values for these parameters. In this talk, we develop a method,
Optimizing the Objective Function Method, to estimate the
multi-parameters. By using dynamical model and the limited data
available, we can obtain accurate approximation of the parameters.